Show that `lim_(xrarr2) ([x-2])/(x-2)` does not exist.

Explanation

We know that
`|x-2|={(x-2),xge2 -(x-2),xlt2`
`therefore R.H.L.=underset(xrarr2)"lim".f(x) (| x-2|)/(x-2)`
`=underset(xrarr2^(+))"lim"(|x-2|)/((x-2))=underset(xrarr2^(+))"lim"((x-2))/((x-2))`
L.H.L. `=underset(xrarr2^(-))"lim"f(x)`
`=underset(Xrarr2^(-))"lim"(|(x-2)|)/(x-2)`
`=underset(xrarr2^(-))"lim"(|x-2|)/(x-2)`
`=underset(xrarr2^(-))"lim"(-(x-2))/(x-2)`
`=-1`
`because R.H.L.neL.H.L`.
`therefore underset(xrarr2)f(x)=underset(Xrarr2)"lim"(|x-2|)/((x-2))` does not exist.